Thank you for providing context, which is always appreciated.
Your PDF confirms that I was correct in my understanding of what you meant.
You have posted this in the wrong forum, this forum is for fluid power not fluid dynamics.
What you are calling "free surface" is part of the phenomenon that we commonly call "open channel flow" in engineering, which is studied in the field of fluid dynamics. I hope that that gives you a good place to start from.
Cheers,
GSTP
Graduate Mechanical Design Engineer
UK